diff --git a/nevanlinna/cauchyRiemann.lean b/nevanlinna/cauchyRiemann.lean new file mode 100644 index 0000000..5d649ed --- /dev/null +++ b/nevanlinna/cauchyRiemann.lean @@ -0,0 +1,50 @@ +import Mathlib.Analysis.Complex.CauchyIntegral +import Mathlib.Analysis.Complex.Basic +import Mathlib.Analysis.Calculus.LineDeriv.Basic +import Mathlib.Analysis.Calculus.ContDiff.Defs +import Mathlib.Analysis.Calculus.FDeriv.Basic +import Mathlib.Analysis.Complex.RealDeriv +import Mathlib.Analysis.Calculus.ContDiff.Basic +import Mathlib.Analysis.Calculus.Deriv.Linear +import Mathlib.Analysis.Complex.Conformal +import Mathlib.Analysis.Calculus.Conformal.NormedSpace +import Mathlib.Analysis.Calculus.FDeriv.Basic +import Mathlib.Analysis.Calculus.FDeriv.RestrictScalars + +variable {z : ℂ} {f : ℂ → ℂ} + +example (h : DifferentiableAt ℂ f z) : f z = 0 := by + + let A := fderiv ℂ f z + let B := fderiv ℝ f + + let C : HasFDerivAt f (ContinuousLinearMap.restrictScalars ℝ (fderiv ℂ f z)) z := h.hasFDerivAt.restrictScalars ℝ + let D := ContinuousLinearMap.restrictScalars ℝ (fderiv ℂ f z) + let E := D 1 + let F := D Complex.I + + have : A (Complex.I • 1) = Complex.I • (A 1) := by + exact ContinuousLinearMap.map_smul_of_tower A Complex.I 1 + + let AR := (ContinuousLinearMap.restrictScalars ℝ (fderiv ℂ f z)) + have : AR (Complex.I • 1) = Complex.I • (AR 1) := by + exact this + + let f₂ := fun x ↦ lineDeriv ℝ f x ⟨0,1⟩ + have : lineDeriv ℝ f z Complex.I = (fderiv ℝ f z) Complex.I := by + apply DifferentiableAt.lineDeriv_eq_fderiv + apply h.restrictScalars ℝ + + have : D Complex.I = Complex.I * (D 1) := by + -- x + + sorry + + have : HasFDerivAt f A z := by + exact DifferentiableAt.hasFDerivAt h + + have : HasFDerivAt f (B z) z := by + sorry + + + sorry diff --git a/nevanlinna/complexHarmonic.lean b/nevanlinna/complexHarmonic.lean new file mode 100644 index 0000000..e69de29 diff --git a/nevanlinna/realHarmonic.lean b/nevanlinna/realHarmonic.lean index 6582dc0..b7889ae 100644 --- a/nevanlinna/realHarmonic.lean +++ b/nevanlinna/realHarmonic.lean @@ -1,10 +1,16 @@ import Mathlib.Analysis.Calculus.LineDeriv.Basic +import Mathlib.Analysis.Calculus.ContDiff.Defs -noncomputable def Real.laplace : (R [× n] → ℝ) → (ℂ → ℝ) := by +noncomputable def Real.laplace : (ℝ × ℝ → ℝ) → (ℝ × ℝ → ℝ) := by intro f - let f₁ := fun x ↦ lineDeriv ℝ f x 1 - let f₁₁ := fun x ↦ lineDeriv ℝ f₁ x 1 - let f₂ := fun x ↦ lineDeriv ℝ f x Complex.I - let f₂₂ := fun x ↦ lineDeriv ℝ f₂ x Complex.I + let f₁ := fun x ↦ lineDeriv ℝ f x ⟨1,0⟩ + let f₁₁ := fun x ↦ lineDeriv ℝ f₁ x ⟨1,0⟩ + let f₂ := fun x ↦ lineDeriv ℝ f x ⟨0,1⟩ + let f₂₂ := fun x ↦ lineDeriv ℝ f₂ x ⟨0,1⟩ + exact f₁₁ + f₂₂ + +structure Harmonic (f : ℝ × ℝ → ℝ) : Prop where + ker_Laplace : ∀ x, Real.laplace f x = 0 + cont_Diff : ContDiff ℝ 2 f